Twenty-four hours and 540 metres separated the gods and the worshippers who gathered before them, giving praise and thanks. As if a miracle had been performed. De Madrid al cielo, they say, and the headlines inevitably ran with the refrain. From Madrid to heaven. Well, via Lisbon.

The first of the faithful arrived at the statue of Cibeles, goddess of fertility, at around 11pm on Tuesday night. It is here that Real Madrid celebrate triumphs and the crowd grew steadily, cars driving past with flags hanging out of windows, horns beeping.

The second congregation arrived at the statue of Neptune, god of the sea, at around 11pm on Wednesday night. It is here, a four-minute walk down the Paseo del Prado from Cibeles, that Atlético Madrid celebrate their triumphs and the crowd grew even more steadily. They have been here rather a lot lately. Soon there were hundreds of them and the songs carried on long into the night. This is historic.

For the first time, two teams from the same city will face each other in the European Cup final. The A5 motorway runs all the way there. It gets called the Carretera de Extremadura, but it is the Road to Lisbon, and they will travel it together, 60 or 70,000 of them. The Stadium of Light is 650km away, around six hours by car. There may never have been a sporting journey like it. Madrid, the capital of European football, packed up and transported to Portugal.

Neither team have won anything yet and it is never wise to celebrate victory before it is achieved. Defeat will be devastating for the loser in the final. Yet no one could begrudge those who came out on to the streets in celebration; this may not be the ultimate success yet but it is a success, cause for joy, for both teams. They have waited a long time for this.

Real Madrid were last in a European Cup final in 2002 when Zinedine Zidane scored that volley at Hampden Park. No team has been as synonymous with a trophy as they have with the European Cup, and winning ladécima, their 10th, has become an obsession. Twelve years has been too long to wait, even if 32 years had passed between the sixth and the seventh successes, the wait punctured by a defeat to Liverpool in the final in 1981.

Since the turn of the century Real Madrid have spent over a billion euros on players. Their manager that night in Glasgow was Vicente del Bosque. He was effectively sacked at the end of the following season, winner of two league titles and two European Cups in four years. Carlos Queiroz, José Antonio Camacho, Mariano García Remón, Vanderlei Luxemburgo, Juan Ramón López Caro, Fabio Capello, Bernd Schuster, Juande Ramos, Manuel Pellegrini and José Mourinho followed him. None of them got there. Now Carlo Ancelotti has.

Atlético Madrid have waited even longer. For them it was never an obsession because it never even felt like a possibility. They have been in only one European Cup final, and that was 40 years ago. Luis Aragonés scored the first in the 114th minute but a ridiculous long-range shot from Bayern Munich's Hans-Georg Schwarzenbeck equalised in the last minute and they lost the replay 4-0. Atlético's goalkeeper was Pepe Reina's dad. It is not just that they have not been back since, it is that they have never even got close.

Atlético's relationship with managers has been tumultuous too but Diego Simeone has changed that. He has changed everything. That night 40 years ago the legend of the pupas, the jinxed one, was born. Over the years it deepened – in 1999 a first ever relegation just three years after winning a league and cup double appeared to be the ultimate expression – and became an excuse with which to justify failure. Simeone, who played for the club, embraced its identity ... but not that part. He will never accept failure.

His has been an extraordinary achievement. Here's a question: has a manager ever had as big and immediate an impact on a team? Brian Clough perhaps?

When Simeone took over in 2011, Atlético hovered just above the relegation zone. They had won the Europa League 18 months earlier, sure, but crisis gripped and they had been knocked out of the Copa del Rey by the second division B side Albacete. Six of the players who lost to Albacete started on Wednesday; a seventh, captain Gabi, was suspended. "I don't want to say 'sunk', but we were in a bad way," Koke recalls. "Simeone changed our life."

In his first (half) season, Atlético won the Europa League and climbed to fifth. In his second, they won the European Super Cup, hammering Chelsea, and finished third, returning to the Champions League. The season ended with them winning the Copa del Rey against Real Madrid – at the Santiago Bernabéu. It was the first time Atlético had beaten their city rivals in 14 years and a fatalism had taken hold. They beat them again at the start of this season. From no wins in 25 to two in two. What happened next was that Radamel Falcao departed. But still they grew.

Although Real knocked Atlético out of the cup this year, the city's "other" team are two wins in three games away from winning a first league title since 1996, a league title that appeared so utterly impossible that even their manager said it could not be won, and they are in the final of the European Cup. "This is madness," Tiago said. No one expected this, even if privately the players did think that a good European run was possible. More possible, in fact, than the league. Amazingly, they could now win both.

On Wednesday no one could doubt their superiority. Even Chelsea's goal was an Atlético goal. Atlético have reached the final without losing a game. They have conceded just once in four matches against Barcelona and Chelsea.

All this with a budget less than a quarter the size of Real Madrid's. It is not as if reinforcements have arrived either. Simeone's starting XI on Wednesday did not include a single new signing. It cost a total of €38.3m in transfer fees. Again, the likely consequence is sales. Atlético, Chelsea and the agents involved all agree on Diego Costa moving to Chelsea. At the moment, the only person who does not is Costa himself. He will not find a club, a life, like this elsewhere. "We might not have the economic might of other clubs but no one can match us for ambition," Simeone said.

At the end of Wednesday night's press conference, Simeone had a message to deliver. "I want to say thanks to the mothers who gave birth to these players: they have huge balls." Together, they have done something barely believable. When they arrived back at Barajas airport it was 5.30 in the morning. Around 100 fans had gathered to greet them, some holding the first edition of the morning's papers. They chanted and sang.

Atlético's hymn was among the songs. It starts with the line: "I'm going to Manzanares." The Manzanares is the river that runs through Madrid, alongside the Vicente Calderón and into the Tagus, heading west all the way to Lisbon.